With the motivation of quantum spatial search, one is interested in the behavior of the quantum walks on various spaces. There are two forms of quantum walks, continuous-time quantum walks and discrete-time quantum walks have been widely studied. In this paper, we restrict our discussion to the discrete time.
Discrete time quantum walks for searching algorithms have been widely studied on finite graphs. However, the limiting distributions of a quantum random walks have only been studied mostly on the line \(\mathbb {Z}\)  using combinatorial methods and recursive methods [1]}, or higher dimensional space \(\mathbb {Z}^d\)  using Fourier transforms [2]}.
It is known that \(d_w=1\) , for standard (moving) Hadamard quantum random walks on \(Z^d\) , for all \(d\) . Yang et al. [3]} used path integral to derive a recursive formula to solve the return probability in \(\mathbb {Z}^d\) . For partially decoherent quantum random walks on \(Z^1\)  with both position- and coin-space decoherence, Zhang [4]} showed that the scaling limit is Gaussian. For partially, only coin-space decoherent quantum random walks on \(Z^1\) , Fan et al. [5]} showed that the distribution of the scaling limit is a continuous
convex combination of normal distributions (but not Gaussian) for quantum random walks in \(\mathbb {Z}\) . For a comprehensive review of quantum walks, see Ref. 2012 review. More recently,
Attal et al. [6]} showed the quantum version of central limit theorem for open quantum random walks. Subsequently, Xiong and Yang [7]} extended their results and showed the scaling limit of a partially open quantum random walk converges to a convex combination of Gaussian distributions.
